Anisotropy

2.23) where a is the distance between the spins, A is called the exchange stiffness constant having

2.4. Anisotropy

When a physical property of any material depends on the direction of the materials that

property is said to exhibit anisotropy. In magnetism, the preference of magnetization to lie

in a particular direction of the sample is called as magnetic anisotropy. As anisotropy plays

an important role in tuning the nature of M–H loop, it is much essential to understand the

various possible sources of the magnetic anisotropy and its influence on the control of the

magnetic properties. Figure 2.06 displays a typical situation where for zero applied field,

the magnetization would point along the easy axis shown ( = 0). When a field is applied,

the magnetization is pulled towards the field direction and approaches closer to the field direction with increasing the applied field. For any intermediate values of

, the

magnetization is being attracted in opposite directions, i.e., up by the field and down by the anisotropy.

Let us assume that all the magnetization is pointing in the same direction in a magnetic material and the material exhibits an easy axis of magnetization. In such scenario, we can describe the energy per unit volume of the magnetization of this material by

𝐸 = 𝐾 𝑠𝑖𝑛²𝛼

(2.24)

where K is called anisotropy constant with a unit of energy per unit volume (J/m

³

or ergs/cc).

Hence, the energy term,

E, is also energy per unit volume. In general, the magnitude of

uniaxial anisotropy is described in terms of the anisotropy field, which is defined as the field needed to saturate the magnetization of a uniaxial crystal in the hard axis direction, as given in eqn.(2.25)

𝐻_𝑘 = 2𝐾_𝑈

𝜇₀𝑀

(2.25)

In general, the energy of the magnetization is given by,

𝐸 = 𝐾 𝑠𝑖𝑛²𝛼 − 𝜇₀𝑀𝐻 𝑐𝑜𝑠(𝛽 − 𝛼)

(2.26) where the first term is anisotropy energy. The second term is due to the magnetic field and the difference in the angle ( - ) is the angle between H and M. In order to get equilibrium, the first derivative is required to be zero.

Figure 2.07: Magnetization of single crystals of iron, nickel and cobalt [HTTP2].

Therefore, taking derivative of eqn.(2.26) with respect to the angle provides,

𝑑𝐸

𝑑𝛼 = 2𝐾 sin 𝛼 cos 𝛼 − 𝜇₀𝑀𝐻 𝑠𝑖𝑛(𝛽 − 𝛼) = 0

(2.27) Taking the value of  as 90

^o

for the equilibrium angle for the magnetization relative to the easy axis and considering the eqn.(2.25) gives

sin 𝛼 = 𝐻

𝐻_𝑘

(2.28)

The above eqn. indicates that when field is zero, the magnetization points along the easy axis and when the field is equal to H

k

, the magnetization points along the direction of field.

For any intermediate value of the applied field, the magnetization points at a value of angle given by eqn.(2.28) rotating smoothly between the easy axis and the applied field.

Figure 2.08: Schematic drawing of broadside and head-to-tail configurations for a pair of ferromagnetically coupled magnetic moments.

2.4.1. Magnetocrystalline anisotropy

Figure 2.07 depicts initial magnetization curves of single crystals of different 3d ferromagnetic elements. It is seen that the materials approach to saturation in different ways when magnetized in different directions. For example, iron display a 100 as easy directions and

111 as hard directions, while nickel exhibits 111 as easy axis and 100 as hard

directions. This behaviour can be understood by analyzing the development of anisotropy energy in different symmetries as given below:

For Hexagonal:

𝐸_𝑎 = 𝐾₁sin²𝜃 + 𝐾₂sin⁴𝜃 + 𝐾₃sin⁶𝜃 + 𝐾₃^′sin⁶𝜃 sin 6𝜙

(2.29) For Tetragonal:

𝐸_𝑎 = 𝐾₁sin²𝜃 + 𝐾₂sin⁴𝜃 + 𝐾₂^′sin⁴𝜃 cos 4𝜙 + 𝐾₃sin⁶𝜃

+ 𝐾₃^′sin⁶𝜃 sin 6𝜙

(2.30)

For Cubic:

𝐸_𝑎 = 𝐾_1𝑐(𝛼₁²𝛼₂²+ 𝛼₂²𝛼₃²+ 𝛼₃²𝛼₁²) + 𝐾_2𝑐(𝛼₁²𝛼₂²𝛼₃²)

(2.31)

where

i

are the direction cosines of the magnetization.

K1c

term is equivalent to

𝐾_1𝑐(sin²𝜃 𝑐𝑜𝑠²𝜙 sin²𝜙 + cos²𝜃 sin²𝜃). When, 

= 0,



= 0, this term reduces to eqn.(2.24) [COEY2010].

Origin of magnetocrystalline anisotropy: There are two distinct sources of

magnetocrystalline anisotropy: (i) single-ion contributions and (ii) two-ion contributions.

The first one is essentially due to the electrostatic interaction of the orbitals containing the magnetic electrons with the potential created at the atomic site by the rest of the crystal.

This crystal field interaction stabilizes a particular orbital and by spin-orbit interaction, the magnetic moment is aligned in a particular crystallographic direction. For example, a uniaxial crystal having 210

²⁸

ions/m

³

described by a spin Hamiltonian DS

²

with D/k

B

= 1 K and S = 2 will have anisotropy constant K

1

= nDS

²

= 1.110

⁶

J/m

³

. On the other hand, the later contribution replicates the anisotropy of the dipole-dipole interaction. Considering the broadside and head-to-tail configurations of two dipoles each with moment m, as shown in Figure 2.08, the energy of the head-to-tail configuration is lower by

3𝜇₀𝑚²/(4𝜋𝑟³) and

hence the magnets tend to align head-to-tail. In non-cubic lattices, the dipole interaction is an appreciable source of ferromagnetic anisotropy.

Figure 2.09: Magnetization of a prolate ellipsoid of revolution with

c > a and no

magnetocrystalline anisotropy. c-axis is the easy direction of magnetization.

2.4.2. Shape anisotropy

Shape anisotropy arising due to the asymmetric shape of the material is important for thin

films where one dimension is limited as compared to other two dimensions. The

demagnetization field inside the material or the stray field outside the magnetic material

depends on the magnetization and shape of the material [JILE1997, OHAN2000,

BLUN2001]. The magnetostatic energy of a ferromagnetic ellipsoid (see Figure 2.09) with

magnetization M

S

is given as

𝐸_𝑚 =1

2𝜇₀𝑉𝑁𝑀_𝑆²

(2.32)

The anisotropy energy is related to the difference in energy

E when the ellipsoid is

magnetized along its hard and easy directions. N is the demagnetization factor tensor for the easy direction. N=(1/2)(1-N) is the demagnetization factor tensor for the hard directions.

Hence,

∆𝐸_𝑚 =1

2𝜇₀𝑉𝑀_𝑆²[1

2(1 − 𝑁) − 𝑁]

∆𝐸_𝑚 = 1

4𝜇₀𝑉𝑀_𝑆²[1 − 3𝑁]

𝐾_𝑠ℎ = 1

4𝜇₀𝑀_𝑆²[1 − 3𝑁]

(2.33)

Table 2.01: Demagnetization factors (in Gaussian units) of selected shapes:

Shape

N1 N2 N3

Sphere 4/3 4/3 4/3

Long Cylinder along z-axis 2 2 0

Infinite plate normal to z-axis 0 0 4

Strip film normal to z-axis

(with t – thickness, W – Width, L – Length; t  W  L)

0 8t/W 4

In addition, the demagnetization factor tensor that relates the demagnetization field with a specimen magnetization as a function of position is given by [NEAL1994]

𝑁(𝑟) = − 1

4𝜋∭ 𝑑³𝑟′∇′ (∇′ ( 1

𝑟 − 𝑟′))

(2.34)

This tensor is given by an integral over the object volume and can be evaluated either inside or exterior to the body. The value of tensor N significantly depends on the specimen shape, which is difficult to obtain in closed-form. It may be calculated exactly for an ellipsoidal shape only. In many symmetrical materials such as any ellipsoid of revolution, the demagnetization factor tensor only has three principal components, i.e.,

( 𝐻₁ 𝐻₂

𝐻₃) = − (

𝑁₁ 0 0 0 𝑁₂ 0 0 0 𝑁₃) (

𝑀₁ 𝑀₂

𝑀₃)

(2.35)

where

N1

+ N

2

+ N

3

= 1 (in SI) and

N1

+ N

2

+ N

3

= 4 (Gaussian). The demagnetization factors for the selected shapes are summarized in Table 2.01. A detailed calculation of demagnetization factor for various objects can be found in [NEAL1994]. The shape anisotropy is zero for a sphere, as

N = 1/3. Shape anisotropy is fully effective in samples

which are so small that they do not break up into domains [COEY2010].

Figure 2.10: Magnetization of a thin film with induced anisotropy created by annealing in a magnetic field. The sheared (open) loop in a (b) is observed when the measuring field H is applied perpendicular (parallel) to the annealing field direction.

2.4.3. Induced anisotropy

In some materials, the magnetic anisotropy can be induced by many ways: (i) fabricate a film in the presence of a magnetic field, (ii) post annealing the materials in the presence of magnetic field and (iii) apply uniaxial stress. In the first two cases, after such treatment, the material may exhibit an easy axis of magnetization that points in the direction of the magnetic field. This induced anisotropy is certainly independent of any crystalline anisotropy or any other form of anisotropy that might be present. Figure 2.10 shows the typical example of inducing the anisotropy in ferromagnetic materials by field annealing.

In the last case, the uniaxial anisotropy is induced by applying uniaxial stress () in a ferromagnetic solid [KRON2003]. The magnitude of the stress-induced anisotropy is

𝐾_𝑢𝜎 =3

2𝜎𝜆_𝑆

(2.36)

where

S

is the saturation magnetostriction. Both the single-ion and two-ion anisotropy

contribute to the stress induced anisotropy. The highest values of uniaxial anisotropy are

found in hexagonal and other uniaxial crystals. Smallest values are found in cubic alloys and amorphous ferromagnets.

Figure 2.11: Schematic drawing of bars to demonstrate inducing an easy-axis in a material with the positive magnetostriction.

2.4.4. Magnetostrictive anisotropy

Another important form of anisotropy in magnetic materials is due to magnetostriction, a change of volume of an isotropic crystal due to magnetic order. Magnetostriction relates the stress in a magnetic material to an anisotropy created by that stress. Figure 2.11 shows the schematic views of bars with different applied stress conditions. If

 is positive, then

application of a tensile stress to the bar creates an easy axis in the direction of the applied stress. If a compressive stress is applied, then the direction of the easy axis created will be perpendicular to the stress direction. On the other hand, if the magnetostriction constant for the material is negative, then the above phenomena would be reversed: a tensile stress will create an easy axis perpendicular to the stress direction and a compressive stress will create an easy axis in the direction of the applied stress.

2.4.5. Magnetic surface anisotropy

The orientation of magnetic moments in the ultra-thin film strongly affects the magnetic

properties and hence attracts enormous interest in magnetic based random access memory

and recording industry since past decades [PAND2016, IKHT2018]. It has been reported

that in ultra-thin films, magnetic moments orient along perpendicular direction of the film

plane up to certain critical thickness due to the magnetic surface anisotropy, which has the following origins: a) the lack of the neighbors or reduced symmetry at the interface gives rise to magnetocrystalline surface anisotropy [NEEL1954], b) strain at the interface due to the lattice mismatch between the substrate and the film and c) the interface roughness. The direction of magnetization in the ultra-thin films is determined by the competition between shape (or dipolar anisotropy) and the magnetic surface anisotropy, i.e., when the thickness of the films is below the critical thickness, the surface anisotropy dominates over the shape anisotropy. On the other hand, the shape anisotropy dominates over the surface anisotropy above critical thickness, which leads to in-plane orientation of the magnetization [PESC1987, GARR2005, YILD20091, YILD20092, HIND2011]. Critical thickness and the easy direction of uniaxial surface anisotropy depend strongly on the deposition conditions, nature of substrate, temperature, materials and microstructure of the film [BRUN1989]. For example, the magnetization of Ni [111] ultrathin films deposited in Au/Ni/Au [111]

multilayers always lies within the plane of films [BRUN1989]. On the other hand, the easy axis of magnetic surface anisotropy of body centered cubic (bcc) Fe [100] grown on Ag [100] is normal to the film plane [HEIN1987, KOON1987]. Hexagonal closed packed (HCP) Co deposited at room temperature on atomically flat polycrystalline Au[111] and covered by Au exhibits perpendicular magnetization for thickness less than 12 Å (6 ML) [CHAP1988]. Recently, CoFeB films show strong thickness dependent magnetic properties due to various anisotropy contribution including surface and interfacial anisotropy [NAIK2012, LIUT2012].

Figure 2.12: Schematic of 90, 180 and domain image of 360 [ZHAN2016] domain walls

in materials.